Lesson 12

Proofs about Quadrilaterals

  • Let’s prove theorems about quadrilaterals and their diagonals.

Problem 1

Lin is using the diagram to prove the statement, “If a parallelogram has one right angle, it is a rectangle.” Given that \(EFGH\) is a parallelogram and angle \(HEF\) is right, which reasoning about angles will help her prove that angle \(FGH\) is also a right angle?

Quadrilateral EFGH with angle HEF as a right angle. Line segments EG and HF intersect at point D.
A:

Corresponding angles are congruent when parallel lines are cut by a transversal.

B:

Opposite angles in a parallelogram are congruent.

C:

Vertical angles are congruent.

D:

The base angles of an isosceles triangle are congruent.

Problem 2

\(ABDE\) is an isosceles trapezoid. Select all pairs of congruent triangles. 

Isosceles trapezoid A B D E. Angles are marked with tick marks, as follows: D E A, one mark; E A B, two marks; A B D, two tick marks; B D E, one tick mark. Segment A E and B D each have one tick mark.  

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A:

Triangle \(ABE\) and triangle \(DBE\)

B:

Triangle \(ABD\) and triangle \(DAE\)

C:

Triangle \(ABE\) and triangle \(BAD\)

D:

Triangle \(AED\) and triangle \(BDE\)

E:

Triangle \(EAB\) and triangle \(EDB\)

Problem 3

Match each conjecture with the rephrased statement of proof connected to the diagram.

Quadrilateral EFGH. Line segments EG and HF intersect at point K.
A:

The diagonals of a parallelogram bisect each other.

B:

In a parallelogram, opposite sides are congruent.

C:

A quadrilateral with opposite sides congruent is a parallelogram.

D:

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

1:

In quadrilateral \(EFGH\) with \(GH\) congruent to \(FE\) and \(EH\) congruent to \(FG\), show \(EFGH\) is a parallelogram.

2:

In parallelogram \(EFGH\), show \(GH\) is congruent to \(FE\) and \(EH\) congruent to \(FG\).

3:

In quadrilateral \(EFGH\) with \(EK\) congruent to \(KG\) and \(FK\) congruent to \(KH\), show \(EFGH\) is a parallelogram.

4:

In parallelogram \(EFGH\), show \(EK\) is congruent to \(KG\) and \(FK\) congruent to \(KH\).

Problem 4

Which of the following criteria always proves triangles congruent? Select all that apply. 

A:

Corresponding congruent Angle-Side-Angle

B:

Corresponding congruent Side-Angle-Side 

C:

Corresponding congruent Side-Side-Angle

D:

3 congruent sides

E:

2 congruent sides

F:

3 congruent angles

(From Unit 2, Lesson 11.)

Problem 5

Select all true statements based on the diagram.

Quadrilateral ABCD. Line AB is parallel to line DC, both cut by congruent transversals AD and BC. Diagonals AC and DB intersect at point E.
A:

Segment \(EB\) is congruent to segment \(AD\).

B:

Segment \(DC\) is congruent to segment \(AB\).

C:

Segment \(DA\) is congruent to segment \(CB\).

D:

Angle \(CBE\) is congruent to angle \(ABE\).

E:

Angle \(CEB\) is congruent to angle \(DEA\).

F:

Line \(DA\) is parallel to line \(CB\).

G:

Line \(DC\) is parallel to line \(AB\).

(From Unit 2, Lesson 10.)

Problem 6

Diego states that diagonal \(WY\) bisects angles \(ZWX\) and \(ZYX\). Is he correct? Explain your reasoning.

Kite W X Y Z. Diagonal W Y is drawn in. Sides Z W and W X have double tick marks. Sides X Y and Y Z have single tick marks.
(From Unit 2, Lesson 9.)

Problem 7

Sketch the unique triangles that can be made with angle measures \(80^{\circ}\) and \(20^{\circ}\) and side length 5. How do you know you have sketched all possibilities?

(From Unit 2, Lesson 4.)